If-0-x-f-t-dt-x-x-1-tf-t-dt-then-find-the-value-of-f-1-

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Question Number 261 by raj last updated on 25/Jan/15

$$\mathrm{If}\underset{0}{\stackrel{x}{\int }}f\left(t\right)dt=x+\underset{x}{\stackrel{1}{\int }}tf\left(t\right)dt,\mathrm{then}\mathrm{find}\mathrm{the}\mathrm{value}$$$$\mathrm{of}f\left(1\right).$$

Answered by prakash jain last updated on 17/Dec/14

$${\int }_0^{x}f\left(t\right)dt=x+{\int }_x^{1}tf\left(t\right)dt$$$$\frac{d}{dx}{\int }_0^{x}f\left(t\right)dt=\frac{d}{dx}[x+{\int }_x^{1}tf\left(t\right)dt]$$$$f\left(x\right)=1+0-xf\left(x\right)$$$$f\left(x\right)=\frac{1}{1+x}$$$$f\left(1\right)=\frac{1}{2}$$

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